lagrange multiplier: dimensions of cylindrical part of tank

[cheffy]

You are designing a gas tank using the smallest amount of material possible. The tank is cylindrical with hemispherical ends and it is to hold 8000 m^3 of gas. What radius and height do you recommend for the cylindrical portion of the tank?

I'm confused with how I would set up my formulas. One I'm assuming would be for a cylinder like x^2+y^2=1 and another would be equal to 8000? How does the hemispherical ends play into this problem?

Thanks!

 

[galactus]

You have a cylinder with surface area \(\displaystyle \L\\2{\pi}rh\) and volume \(\displaystyle \L\\{\pi}r^{2}h\)

The surface area of a sphere is given by \(\displaystyle \L\\4{\pi}r^{2}\) with volume \(\displaystyle \L\\\frac{4}{3}{\pi}r^{3}\)

The two hemispheres make up a sphere.

So you have the total volume of the tank as:

\(\displaystyle \L\\V={\pi}r^{2}h+\frac{4}{3}{\pi}r^{3}=8000\)

The surface area is:

\(\displaystyle \L\\S=2{\pi}rh+4{\pi}r^{2}\)

S is what you must minimize. Solve the volume equation for, say, h and sub it into the surface equation. Then you can differentiate, set to 0 and solve.

 

[soroban]

Re: lagrange multiplier: dimensions of cylindrical part of t

Hello, cheffy!

You are designing a gas tank using the smallest amount of material possible.
The tank is cylindrical with hemispherical ends and it is to hold 8000 m^3 of gas.
What radius and height do you recommend for the cylindrical portion of the tank?

Galactus is absolutely correct . . .

The surface area is: \(\displaystyle \:S \;=\;2\pi rh\,+\,4\pi r^2\)

The constraint is: \(\displaystyle \:V \:=\:\pi r^2h\,+\,\frac{4}{3}\pi r^3 \:=\:8000 \;\;\Rightarrow\;\;\pi r^2h\,+\,\frac{4}{3}\pi r^3\,-\,8000\:=\:0\)


Using Lagrange Multipliers, we have:

. . \(\displaystyle F(r,h,\lambda) \;=\;2\pi rh\,+\,4\pi r^2 \,+\,\lambda\left(\pi r^2h\,+\,\frac{4}{3}\pi r^3\,-\,8000\right)\)


Now solve the system: \(\displaystyle \:\begin{array}{ccc}\frac{\partial F}{\partial x} & = & 0 \\ \\ \\ \frac{\partial F}{\partial h} & = & 0 \\ \\ \\ \frac{\partial F}{\partial\lambda} & = & 0 \end{array}\)